Compatibility and attainability of matrices of correlation-based measures of concordance

TL;DR

This paper characterizes a broad class of correlation-based concordance measures, studies when matrices of these measures can be realized by random vectors, and proposes methods for constructing such vectors, especially for complex block matrices.

Contribution

It introduces a unified framework for correlation-based concordance measures, analyzes their matrix compatibility and attainability, and develops new methods for constructing random vectors with specified measures.

Findings

Characterization of a broad class of concordance measures including Spearman's rho and Blomqvist's beta.

Conditions for the compatibility and attainability of matrices of these measures.

A new subclass of attainable block Spearman's rho matrices for higher dimensions.

Abstract

Measures of concordance have been widely used in insurance and risk management to summarize non-linear dependence among risks modeled by random variables, which Pearson's correlation coefficient cannot capture. However, popular measures of concordance, such as Spearman's rho and Blomqvist's beta, appear as classical correlations of transformed random variables. We characterize a whole class of such concordance measures arising from correlations of transformed random variables, which includes Spearman's rho, Blomqvist's beta and van der Waerden's coefficient as special cases. Compatibility and attainability of square matrices with entries given by such measures are studied, that is, whether a given square matrix of such measures of concordance can be realized for some random vector and how such a random vector can be constructed. Compatibility and attainability of block matrices and hierarchical matrices are also studied due to their practical importance in insurance and risk management. In particular, a subclass of attainable block Spearman's rho matrices is proposed to compensate for the drawback that Spearman's rho matrices are in general not attainable for dimensions larger than four. Another result concerns a novel analytical form of the Cholesky factor of block matrices which allows one, for example, to construct random vectors with given block matrices of van der Waerden's coefficient.